The Yang-Mills heat equation with finite action

Leonard Gross

Published 2016-06-13Version 1

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over three dimensional Euclidean space and over a bounded open convex set therein. The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by using the Zwanziger-Donaldson-Sadun method. This consists in solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in a gauge group whose informal Lie algebra consists of functions lying in the Sobolev space of index three halves. Since the supremum norm of such functions is not controlled by this Sobolev norm, the nature of this group must be itself understood in order to carry out the reconstruction procedure. Properly defined, this group is shown to be a complete topological group in its natural metric. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.